How π back-donation quantitatively controls the CO stretching response in classical and non-classical metal carbonyl complexes

We definitively show that the CO stretching response to metal coordination is driven exclusively by π polarization, which quantitatively correlates with π back-donation and changes in CO bond length and frequency.


Introduction
The high affinity of carbon monoxide (CO) towards metals (M) has been known since the end of the nineteenth century 1 and its relevance has kept growing thereaer, both in pure 2,3 and applied chemistry. 4,5 This has led many chemists to study in detail the coordination bond between M and CO in metalcarbonyl complexes, which is commonly described in terms of the Dewar-Chatt-Duncanson (DCD) model. [6][7][8] According to this scheme, the interaction between M and CO involves the donation of electron charge from the carbon's lone pair to the empty M orbitals of s symmetry (M ) CO s donation), and a back-donation from lled M to empty CO orbitals of p symmetry (M / CO p back-donation). The effectiveness of this model for the description of the M-CO bond has been consolidated over the years by a large number of theoretical studies based on a variety of techniques, including energy 9,10 and charge 11 decomposition schemes, Natural Bond Orbitals (NBO) analysis, 12 Electron Localization Function (ELF) approaches 13 and the Quantum Theory of Atoms In Molecules (QTAIM). [14][15][16][17] On the experimental side, discussions on the nature of the M-CO bond are mostly based on the analysis of the variation in the CO stretching frequency n CO (via IR spectroscopy) and bond distance r CO (via X-ray crystallography) with respect to free CO (n free-CO ¼ 2143 cm À1 , r free-CO ¼ 1.12822Å). In most metalcarbonyl complexes the CO bond appears weakened, i.e., the stretching frequency decreases (Dn CO ¼ n CO À n free-CO < 0) and the bond distance increases (Dr CO ¼ r CO À r free-CO > 0), but in a minority of (mainly late-metal cationic) complexes, which are sometimes termed "non-classical", 18 the CO bond appears strengthened (Dn CO > 0 and Dr CO < 0). These differences in the CO stretching response to the M-CO bond formation in metal carbonyl complexes are commonly explained in terms of the relative importance of the DCD constituents of the M-CO bond. In particular, M / CO p back-donation is represented as exerting a bond-weakening effect on CO, while M ) CO s donation is thought to act in the opposite way. 19,20 This view relies on a molecular-orbital picture in which both the p acceptor and s donor CO orbitals have a C-O anti-bonding character. However, while there is general agreement on the effect thus played by p back-donation, the role of s donation has been brought into question in the last een years. [21][22][23] In particular, these studies suggest that the s CO donor orbital has, rather, a weak bonding character and that the CO bond strengthening in non-classical complexes is an electrostatic effect due to the (positively charged) ligand-metal moiety, whereby the CO bonding orbitals of both s and p symmetry are polarized in the C ) O direction, thus enhancing the covalency of the CO bond.
One way to schematically depict M(CO) bonding resorts to a simple Valence Bond (VB) picture. Focusing on the M(CO) moiety of a generic [(L) n M(CO)] m complex, three VB structures differing for the extent of p back-donation can be written: In going from structure (a) to structure (b) and (c), where one has zero, one and two p* orbitals of CO engaged in backbonding, the CO bond multiplicity goes from three to two to one. The relative weight of each structure will of course depend on the p donor properties of the specic [(L) n M] m fragment. At the same time, the electronic structure of CO is also affected by the electric eld generated by this fragment, especially in those cases when m s 0. For CO in the presence of an electric eld generated, for instance, by a positively charged metal fragment (exemplied here with the symbol 4), three analogue VB structures can be written: The presence of such electric eld would in this case favour the triple bonded structure (d) over structures (e) and (f) featuring a double and single bond, respectively (an opposite effect, of course, is expected to occur when the electric eld is generated by an anionic ligand-metal fragment). The DCD bonding structure and the electrostatic polarization effect may thus a priori act in different directions with different weight, so that their interplay in driving CO stretching response may be difficult to disentangle.
Still, however, carbonyl complexes showing blue shied (Dn CO > 0) CO stretching frequencies are most oen assumed to lack back-donation from the metallic fragment. 18,24,25 Exemplary in this respect is the set of complexes [(L)Au(CO)] 0/+ of gold(I) that have been experimentally characterized. 24,[26][27][28][29][30][31] Until last year, to our knowledge, nine gold(I) carbonyl complexes had been spectroscopically characterized: the ligand free [Au(CO)] + (observed in neon matrix 26 ) and its derivatives with ligands Cl À , 27 3 ] À is a uorinated tris(pyrazol)borate ligand. They all exhibit blue shi of the CO frequency and therefore are classi-ed as non classical. This has been taken by some authors as proof that the gold fragment gives poor or no back-donation. 32,33 However, in apparent contradiction, both theoretical and experimental studies have shown that the p donor character of gold is usually far from negligible [34][35][36] (especially toward carbon monoxide 37 ) with important effects in catalysis. 38,39 Recently, furthermore, a gold(I) complex showing Dn CO < 0 has been fully characterized, 40 bearing a neutral o-carborane diphosphine (DPCb) as an ancillary ligand. Such an "exception", which is even more singular when considering that the formal positive charge should strengthen the CO bond, made the authors speak of "enhanced p back-donation" from the [(DPCb)Au] + fragment. For the reader's convenience, an overview of the experimentally characterized systems, with the reported Dn CO values and reference to the original papers, is displayed in Fig. 1 35,42 with spectroscopic observables has been the subject of some of our recent work, 36,37,43 and in the present work we have used this analysis to systematically study an extensive series of carbonyl compounds. The unique power of CD analysis lies in the fact that it provides a complete picture, across the entire molecular space, of the charge ow of s and p Fig. 1 The experimentally characterized gold(I) carbonyl complexes discussed in this work, with the observed carbonyl stretching frequency shifts, Dn CO (in cm À1 ), and literature references.
character accompanying the formation of a coordination bond, and it permits a well-dened, consistent measure of the charge transfer (CT) associated both with the DCD components of the M-CO bond and with the s and p components of the polarization 44 taking place at the CO ligand itself. As a result, as we hope we will have convinced the reader by the end of the paper, this work provides a denitive and quantitative account of the role and interplay of the DCD components of the M-CO bond and of CO polarization in driving CO stretching response to coordination.
We thus investigate the relation between Dn CO and Dr CO and the charge displacements of s and p symmetry along the M-C-O axis in response to the M-CO bond formation in metal carbonyl complexes. We carry out our analysis rst on an exhaustive series of 23 gold(I) carbonyls of formula [(L)Au(CO)] 0/+ , where L is a varying auxiliary ligand (including none), which includes 8 of the experimentally characterized complexes and which is evenly partitioned between charged and neutral complexes, as well as between classical (CO bond elongated and frequency red-shied) and non-classical (CO bond shortened and frequency blue-shied). The choice of binary gold complexes seems to be particularly simple and useful, as it permits to isolate and study systematically the effect of the trans ligand across a wide variety of metal binding properties and electronic effects. We begin our analysis (Section 3.1) by studying in greater detail the two extreme cases of "naked" Au + , [Au(CO)] + , which displays the experimentally largest blue-shi, and of [(DPCb)Au(CO)] + , which is the only known case of a positively charged but significantly red-shied gold(I) complex. Having thus highlighted the main ndings, we then thoroughly conrm them by extending the study to the whole series of complexes (Section 3.2). To complete the work we then also investigate the role of the metal itself in driving CO response to coordination, by studying a series of homoleptic [(CO) n M(CO)] m complexes, with M including Hg, Ir, Ni, Fe, Cr, Mo, Co, Ru (Section 3.3). Finally, an ad hoc study of CO in a uniform axial electric eld (Section 3.4) concludes the work, in order to isolate the impact of CO polarization and of its s and p components on CO stretching response.

Methodology and computational details
In the charge-displacement (CD) analysis framework, a chemical bond A-B is analyzed in terms of the difference Dr(x,y,z) between the electron density of the adduct AB and that of the two non-interacting fragments A and B frozen at their in-adduct geometries. A partial progressive integration of Dr(x,y,z) along a suitably chosen bond axis z yields the so called chargedisplacement function (CDF) 42

Dr
x; y; z 0 dx dy: The CDF denes, at each point z, the exact amount of electron charge displaced from right to le (the direction of decreasing z) upon bond formation through a plane perpendicular to the z axis through the point z (negative CDF values indicate a charge ow in the opposite direction). If both the adduct and its constituting fragments have proper symmetry, Dr(x,y,z) can be decomposed into additive components of s and p symmetry with respect to the bond axis z (see ref. 35 for further details).
All of the complexes studied in this work have general formula [(L) n M(CO)] m . Since the M-CO bond is under investigation, the appropriate fragments are the ligand-metal moiety [(L) n M] m and carbon monoxide CO, and the z reference axis joins the M and C centres. For the purpose of separating the s and p components of Dr(x,y,z), we group the orbitals of adduct and fragments according to the irreducible representations of the complex (and fragments) symmetry groups, which in the present cases are either the C 2v group (where the A 1 representation corresponds to s, while B 1 and B 2 correspond to p) or the C 3v group (A 1 corresponding to s donation, and E 1 and E 2 to p back-donation). No CO orbital is of A 2 symmetry, therefore this representation is not relevant for the DCD analysis of the M-CO bond and is found to represent only a (minor) rearrangement internal to the ligand-metal fragment. Among the gold(I)  Table 2. The reduced symmetries C 2v and C 3v have been used to separate the s and p components of the electron density difference also for these complexes.
The CDFs of the s and p components of Dr(x,y,z) provide a thorough, spatially detailed picture of the DCD donation and back-donation charge ows. 35 Well-dened measures of the net charge transfer and of its donation and back-donation contributions (hereaer CT net , CT s don and CT p back , respectively) can be obtained by taking the CDFs values at a plausible inter-fragment boundary, which we take to be the z point where equalvalued isodensity surfaces of the fragments become tangent. 35,36 As mentioned in the Introduction, in the present context the CDF also provides precious additional information concerning CO polarization. Since the C-O bond is collinear with the M-C z axis of integration, the CDF in the C-O bond region represents the electron displacement within CO with respect to free CO in response to the M-CO bond formation. The amount of charge owing across a plane normal to the CO bond through its midpoint (i.e., the CDF value at z ¼ r CO /2) can be usefully taken as a quantitative estimate of such polarization, and the total value can again itself be decomposed in s and p components. We shall refer to these values as to CT r CO /2 , CT s rCO=2 and CT p rCO=2 , respectively.
Geometry optimizations and the calculation of harmonic frequencies and electron densities were carried out by means of Density Functional Theory (DFT) with the ADF package. [45][46][47] Becke's exchange functional 48 in combination with the Lee-Yang-Parr correlation functional 49 (BLYP) was adopted. We used an all electron triple-zeta basis set with two polarization functions (TZ2P) and a small frozen core for all atoms. Relativistic effects were included via the zeroth-order regular approximation (ZORA) Hamiltonian. [50][51][52] An assessment of the effect of the exchange-correlation functional and of the basis set on the CDF is given in the ESI, † where a comparison is also made with results from fully relativistic calculations carried out with a recently implemented parallel version of the Dirac-Kohn-Sham program BERTHA. [53][54][55] The purely electrostatic effect on the CO charge rearrangement was investigated using a uniform axial electric eld (see also ref. [56][57][58] orientated along the C-O bond axis z (more details are given in Section 3.4). The density difference Dr(x,y,z) in this case was formulated as the electron density of CO in the presence of the electric eld at the actual minimum energy conguration minus that of free CO at the same geometry.

Results and discussion
As mentioned in the Introduction, we rst describe here a detailed investigation of the M-CO bond in [Au(CO)] + and [(DPCb)Au(CO)] + (Section 3.1). We then extend the analysis to a whole series of 21 [(L)Au(CO)] 0/+ complexes (Section 3.2) and, nally, to a series of nine homoleptic complexes of general formula [(CO) n M(CO)] m (Section 3.3). The full list of complexes considered is in Tables 1 and 2. The purely electrostatic effect is investigated in the last Section (3.4) where an analysis of CO in a uniform axial electric eld is carried out.
[(DPCb)Au(CO)] + , however, is only slightly asymmetric in its minimum conguration and has been here constrained to C 2v symmetry (the difference in energy with respect to the unconstrained optimized conguration is as small as 1 kcal mol À1 ). The other two have been excluded from our analysis because they are much more asymmetric and to constrain them to C 3v symmetry would probably alter their properties signicantly.
As Table 1 shows, the experimental CO stretching frequency for the three complexes [(CF 3 )Au(CO)], [(Cl)Au(CO)] and [(Br) Au(CO)] (the rst of which is measured in the solid state and the others in solution) is actually blue-shied rather than red-shied as the calculations consistently suggest for all the neutral systems (the computed v free-CO is 2143 cm À1 ). Regarding this apparent inconsistency, Frenking et al. recently found that the experimental blue shi is actually due to the presence of intermolecular interactions and not to the properties of the single molecule. 59 They proved this by computing the CO frequency of small aggregates of [(CF 3 )Au(CO)] and of [(Cl) Au(CO)] and nding that the frequency increases from smaller to larger values than that of free CO. Indeed, Au-Au interactions have been experimentally observed for these two complexes in the solid state 24,60 and are likely to occur also in solution, especially for ligands with little steric hindrance. For this reason, and since experimental data are available only for a small subset of the complexes considered here, we shall base our discussion on the DFT values of Dn CO and Dr CO (computed r free-CO ¼ 1.137Å). In fact, we shall most oen refer Table 1 Computed Dn CO (cm À1 ) and Dr CO (Å) and charge-transfer results (e) obtained from the CD analysis for the considered series of [(L) Au(CO)] 0/+ complexes. In boldface, data for the experimentally observed complexes. Reference values are n free-CO ¼ 2106 cm À1 (experimental: 2143 cm À1 ), r free-CO ¼ 1.137Å. a The vibrational coupling between the CO and the ligand has been eliminated through isotopic substitution ([(C 28  to the latter parameter only, because the non-uniform inuence of vibrational mode coupling, and the more complicated CO vibration modes in the homoleptic carbonyls, make Dn CO a less reliable parameter than Dr CO for a quantitative analysis of its relation with the M-CO bond characteristics.

[Au(CO)] + and [(DPCb)Au(CO)] +
We start our analysis with an in-depth investigation of the gold  Fig. 2 the CDFs for the overall density difference and its symmetry-separated components. We recall here that, at a given point z, a positive CDF value corresponds to a charge ow from right to le (i.e., in the Au + ) CO direction) while a negative value corresponds to a charge ow in the opposite (Au + / CO) direction. The total CDF is positive over both the Au-C and C-O bond regions and also at the oxygen far side of CO, indicating a continuous ow of electrons in the direction from CO towards gold. The negative values of the curve on the le side of Au + indicate a rearrangement in the opposite direction, which was shown in ref. 42 to be due to gold sd hybridization. The total CDF results from an A 1 component which is large and positive in the Au-carbon region (identifying s donation) and a B 1 + B 2 component which is negative in the same zone (identifying p back-donation) plus a negligible A 2 component. These components are easily recognized in the isodensity plots of the respective density difference shown at the top of the gure.
The net charge transfer CT net from CO to Au + (the CDF value at the boundary solid vertical line) amounts to 0.16e resulting from a donation component CT s don of 0.34e and a back-donation component CT p back of 0.18e. The rst important comment here is that, in a system like this showing a large blue-shi of the CO stretching frequency, back-donation is actually a signicant component of the interaction, estimated to be more than half as large as the donation.
An analogous signicant contribution from the electron charge rearrangement of p symmetry was also recently highlighted in ref. 29 through a Natural Orbitals for Chemical Valence-Extended Transition State (NOCV-ETS) 61 energy decomposition analysis. In particular, the p contribution to the overall orbital interaction energy DE orb was found to be surprisingly large (32.5% of the overall DE orb ). The authors were cautious, however, in attributing such contribution exclusively to p back-donation, as DE orb not only accounts for genuine inter-fragment orbital interactions but also for the polarization of the orbitals within each fragment.
This uncertainty may be dissolved here, because, as discussed in Section 2, the interfragment charge transfer and its components are automatically separated from the corresponding components of CO polarization in the CDF picture. Inspection of Fig. 2 is in fact particularly revealing in this respect. Focusing on the CDFs in the carbonyl region, we notice immediately that the positive value of the total function indicates that the CO bond is on the whole polarized in the C ) O direction. Remarkably, this polarization results from the concordant positive contributions of both the s and p components. We indeed see that, while the s CDF keeps its (positive) sign on the right hand side of C and even beyond the oxygen site, an inversion (from negative to positive) is seen to occur for the p component precisely at the carbon site, leading to a maximum located at about the mid-point of the C-O bond. In both cases, therefore, there is a displacement of electrons from oxygen towards carbon, which is due to the presence of the positively charged metal fragment. As discussed in Section 2, we can quantify the extent of CO polarization by taking the CDFs values at the mid-point of the CO bond (dashed vertical line in Fig. 2). For the case under examination, the C ) O polarization amounts to CT r CO /2 ¼ 0.16e, resulting from a s contribution CT s rCO=2 of 0.07e and a p contribution CT p rCO=2 of 0.09. We now turn to [(DPCb)Au(CO)] + , with its CDFs reported in Fig. 3. This is analogous to Fig. 2 except that here the B 1 (dashed blue line) and B 2 (dotted-dashed line) components are not identical and are shown separately in the plot. We notice an immediate striking contrast with the previous [Au(CO)] + case, in that the back-donation components globally dominate over s donation in the coordination bond region, so that the total CDF is negative everywhere, indicating a continuous, though modest, ow of electrons from [(DPCb)Au] + to CO. This conrms the already cited ndings of ref. 40. We note that p back-donation is in turn largely dominated by the B 2 component. The net charge transfer at the inter-fragment boundary is À0.06e, resulting from a s donation component of 0.26e (A 1 ) and a p back-donation component of À0.32e (À0.07 due to the B 1 component and À0.25 due to the B 2 component).
The polarization of the electron cloud in the carbonyl region also differs remarkably from that in [Au(CO)] + . In analogy with [Au(CO)] + , the s CDF remains positive in the CO region and the B 1 component turns positive at the C site, reecting the polarization of the CO bonding orbitals due to the electrostatic effect of the metal fragment. However, by contrast, the B 2 component maintains its negative sign also in the CO region, i.e. the backdonation it represents is so pronounced that it penetrates the CO region and extends even beyond the oxygen. As a consequence, the CO bond is on the whole slightly polarized in the C ) O direction (CT r CO /2 ¼ 0.03e), resulting from a s polarization in the same direction ðCT s rCO=2 ¼ 0:05eÞ and a p polarization in the opposite C / O direction ðCT p rCO=2 ¼ À0:02eÞ. It is worth deepening the comparison between the two complexes examined so far. In both, the metallic fragment bears a formal positive charge. However, [Au(CO)] + behaves nonclassically (blue-shied Dn CO ), while [(DPCb)Au(CO)] + behaves classically (red-shied Dn CO ). The CD analysis reveals that the s donation component of the metal-CO bond is roughly comparable in the two cases (CT s don 0.34 vs. 0.26e), while p backdonation is almost twice as large in [(DPCb)Au(CO)] + (CT p back 0.32 vs. 0.18e) and its extent substantially reduces the C ) O polarization of the CO bond. The polarization of the CO s bonding orbitals is comparable in the two complexes (CT s rCO=2 0.07 vs. 0.05e), but that of the p bonding orbitals is not (CT p rCO=2 0.09 vs. À0.02e). These ndings suggest that p electron displacement upon coordination is the main factor driving CO bond response. In particular, if the presence of the metal fragment, especially if positively charged, is capable of polarizing the p CO bonding orbitals, even in the presence of a signicant back-donation, the CO bond is strengthened; if, on the other hand, p back-donation is strong and extended enough to contrast CO polarization, even in the presence of an equally cationic metal fragment, the CO bond is weakened.

The complete [(L)Au(CO)] 0/+ series
We now need to verify if the above preliminary surmise stands the test of a wider series of carbonyl compounds. To this end, we have extended the analysis to all 23 [(L)Au(CO)] 0/+ complexes listed in Table 1, which collects the spectroscopic data for Dn CO and Dr CO as well as the various computed CT gures. The complexes are listed in order of increasing Dr CO and the experimentally characterized compounds are those shown in boldface. As briey discussed at the beginning of Section 3, it is seen that, according to our computed shis, the neutral complexes plus [(DPCb)Au(CO)] + behave classically, while the remaining cationic complexes behave non-classically. The s donation and p back-donation CDFs for these complexes are collected, respectively, in the top and bottom panel of Fig. 4. Red lines are for the complexes showing red shi of n CO , blue lines are for those showing blue shi.
Two eye-catching features emerge upon inspection of Fig. 4. The rst is that all systems exhibit a surprisingly similar s charge rearrangement (top panel) in the CO fragment region, in contrast with a much wider variability on the metal fragment side and despite the fact that some of the complexes are neutral and some cationic. In fact, as Table 1 shows, if one excludes the special cases of the naked Au + , of the inert ligands Ne and Xe, and of the anomalous [(DPCb)Au(CO)] + , even the net ligand-tometal s donation, CT s don , varies by only 0.05e across the whole series of ligands. On the contrary, the p CDF (bottom panel of Fig. 4) appears to be strongly inuenced by the nature of the ligand over the whole molecular region, and CT p back varies by 0.22e over the ligand series. The second important observation is that, in the CO region, the complexes showing a blue-shied n CO (blue lines) all invariably exhibit a ow of p electrons in the C ) O direction ðCT p rCO=2 . 0Þ, due to the positively charged metallic fragment, while the complexes showing red-shied n CO (red lines) exhibit a negative CT p rCO=2 , i.e., charge ows in the opposite C / O direction (with the exception of two complexes for which CT p rCO=2 is essentially vanishing and the red-shi is also negligibly small).
It thus appears quite clearly that in the series of gold(I) carbonyls: (i) s donation is much less tunable than p back- donation, being very little dependent on the nature and the charge of the ligand; (ii) whereas the net CO bond polarization turns out to be invariably oriented in the C ) O direction (CT r CO /2 > 0), the direction of its p density component can vary and appears to be tightly connected with the direction of the CO stretching shi and bond-length change. These ndings are given a denitive illustration in Fig. 5 and 6 where the correlation of Dr CO with CT net , CT s don , CT p back , CT r CO /2 , CT s rCO=2 and CT p rCO=2 is reported. In both gures, black triangles are used for the overall CT, red squares for its s component and blue circles for its p component. Empty symbols are for the neutral species, lled ones are for the cationic species.
Focusing rst on Fig. 5, no correlation is found, as expected, between Dr CO and CT s don , while a good inverse correlation (R 2 ¼ 0.945) can be seen between Dr CO and CT p back , a trace of which remains in the plot of Dr CO vs. CT net . The same bond weakening effect of p back-donation is also evident in the plot of Dn CO vs.
CT p back (see ESI †), though correlation, as mentioned above, is made worse by mode coupling (R 2 ¼ 0.849). Fig. 6 shows the correlation of Dr CO with CT r CO /2 and its components CT s rCO=2 and CT p rCO=2 . Not surprisingly, as these quantities are all directly related to the charge rearrangement of the CO bond itself, correlations are here quantitatively better (R 2 is 0.970 for that with CT p rCO=2 ). Obviously, as Dr CO correlates well with both p back-donation and CO p electron polarization, the latter two quantities are also in mutual correlation.    the M-CO bond and CO polarization are varied essentially by changing the metal. The full list of the considered homoleptic complexes is in Table 2, reporting their spectroscopic shis and CD decomposition results. Complexes are listed in order of increasing value of Dr CO . We omit for brevity a presentation of the complete CDFs. The computed structures for these systems are in agreement with experimental X-ray data where available. [62][63][64][65][66] Hg(CO) 2 2+ and Ir(CO) 6 3+ , both cationic, behave non classically, with experimental blue-shied n CO at 2279.5 cm À1 for the former and at 2254, 2276 and 2298 cm À1 for the latter. 65,66 On the opposite side, the anionic complexes show exceptionally low CO stretching frequency, the most red-shied being that of Fe(CO) 4 2À at 1730 cm À1 (this is the rst anionic carbonyl complex spectroscopically observed 67,68 ). In between are Mo(CO) 6 , Fe(CO) 5 (for which both the axial and equatorial M-CO bonds have been investigated), 69 Ni(CO) 4 and Cr(CO) 6 . The complexes present therefore a wide range of n CO variation but Dn CO turns out not to be a good parameter for analyzing correlations with the CD data because normal-mode coupling varies signicantly with the different structure of the complexes. We therefore base our discussion, as already done for the gold(I) complexes, on the computed Dr CO . This varies in a range of 0.087Å over the series, from À0.018 to 0.069Å ( Table 2). The table shows that also in this series of compounds the range of variation in p back-donation (0.69e) is much larger than that of s donation (0.15e). In particular, almost no backdonation is found for Hg(CO) 2 2+ while CT p back for [Fe(CO) 4 ] 2À is as high as 0.71e. This picture is consistent with the simple VB view discussed in the Introduction, in that we go from a purely s M-CO bond (structure a) for Hg(CO) 2 2+ to a situation in which all p* CO orbitals are engaged in back-bonding (structure c) for [Fe(CO) 4 ] 2À . Also the charge rearrangement (polarization) in the carbonyl region is seen to follow a similar trend, with a much narrower range of CT s rCO=2 values (between 0.02 and 0.10e) than that of CT p rCO=2 (from 0.21 to À0.30e). As before, no clear correlation can be discerned between Dr CO and the s CT data, while CT p back and CT p rCO=2 values are seen to decrease almost monotonically as Dr CO increases.
A plot of Dr CO vs. either CT p back or CT p rCO=2 for the whole set of complexes studied, including the present homoleptic carbonyls in addition to the gold(I) series, appears in fact to suggest, because the range of variation is now signicantly enlarged, that a quadratic t, rather than a linear one, may better represent the actual correlation (an evident non-linear relationship has already been found between the electric eld strength and Dr CO 56 ). Fig. 7 very clearly shows this to be the case, with the accuracy of all ts improved with respect to the sole subset of gold complexes.
Once again, in the homoleptic series, the carbonyl complexes featuring CO bond strengthening (blue-shied Dn CO and negative Dr CO ), i.e. the cationic Hg(CO) 2 2+ and Ir(CO) 6 3+ , show a ow of p electrons in the C ) O direction. All other complexes, where the CO bond weakens (red-shied Dn CO and positive Dr CO ) show opposite-direction ows.

CO in a uniform axial electric eld
The observation that the CO bond is lengthened or shortened upon formation of the M-CO bond according to whether the CO bonding orbitals of p symmetry are polarized in the C / O or C ) O direction, respectively, is certainly remarkable. To verify that this is a general fact, actually independent of CO coordination, we discuss in this last section an ad hoc study of the electron cloud rearrangement and stretching response of CO in an external uniform axial electric eld oriented along the C-O bond axis. In Fig. 8, we show the computed CO stretching Dr CO reported versus the p and s components of CT r CO /2 . The latter vary as a result of the applied eld in the same gure. The points representing the computed Dr CO and p and s components of CT r CO /2 are reported for the whole series of carbonyl complexes studied in this work.
Let us focus rst on the stretching response to the electric eld. When the eld is absent, the system corresponds to free CO and Dr CO , CT s rCO=2 and CT p rCO=2 are all zero. As the eld increases on the le, in the direction that induces (linearly) C / O (negative) polarization, C-O bond length increases quadratically and p polarization is seen to increase much more rapidly than s polarization. Conversely, as the eld increases on the right, inducing C ) O polarization, the C-O bond shortens (much less rapidly).
When we now compare these curves with the relation observed between Dr CO and the s and p components of CO polarization induced by metal coordination, rather than by an applied eld (disconnected circles in the gure), we notice immediately that the p circles follow quite closely the correlation between eld-induced polarization and stretching, while, in striking contrast, the s circles deviate from the eld-induced line (a clear indication of a much more pronounced "chemical" signature) and, moreover, span a very narrow range of (positive) polarization, essentially without any correlation with the widely varying Dr CO . This is indeed a very strong conrmation that the CO stretching response to any solicitation causing electron charge rearrangement, be it the formation of a M-CO coordination bond or the effect of an external electric eld, is driven essentially exclusively by the charge rearrangement of p symmetry: whether induced by an external electric eld or by metal coordination, C / O (C ) O) polarization of the p bond orbitals invariably and tightly correlates with bond lengthening (shortening).

Conclusions
In this work we have carried out an in-depth analysis of the M-    8 The dashed lines show Dr CO versus the s (red color) and p (blue) CO polarization CT r CO /2 for a CO molecule placed in a uniform axial electric field of magnitude ranging from À0.11 to 0.07 a.u. in steps of 0.02 a.u. (colored square points). For comparison, also shown is the correlation between Dr CO and s and p CO polarization (red and blue circles, respectively) for the whole series of Au and homoleptic complexes studied. The empty circles are for neutral or negatively charged complexes, the filled circles for cationic ones.
donation charges but also of the s and p components of CO polarization were obtained by the well-established chargedisplacement analysis of electron-density differences, as resulting from accurate DFT calculations. The nature of the M-CO bond in the considered complexes was found to range smoothly between the two extreme cases of an almost purely s bonded complex (Hg(CO) 2 2+ , CT p back ¼ 0.02e) and of a strongly back-bonded complex ([Fe(CO) 4 ] 2À , CT p back ¼ 0.71e). Conversely, all complexes were found to feature a narrowly comparable s donation component, with CT s don values ranging from 0.14 to 0.34e. The same picture holds accurately for the electron cloud rearrangement over the carbonyl region: all considered complexes feature a comparable s polarization of CO and a much more variable p polarization. Quite remarkably, no correlation is found between Dr CO and the s displacements, while Dr CO , p back-donation and CO p polarization all correlate tightly with one another.
These results show that the driving force of the CO stretching response to the M-CO bond formation is provided exclusively by the changes taking place in the p electron density. In the complexes studied, such p charge rearrangement is found to result from the interplay between p back-donation (structures a-c of the Introduction) and the electrostatic effect (structures d-f) exerted by the metal-ligand fragment. In particular, cationic metal-ligand fragments polarize the p CO bonding orbitals in the C ) O direction, thus shortening the bond and enhancing the covalency, as highlighted in ref. 23. This effect, on the other hand, is contrasted by p back-donation shiing charge in the opposite direction. The net direction C ) O or C / O of the polarization of p CO bonding orbitals is found to invariably determine whether the CO bond is strengthened or weakened, respectively. This is most evident in the [(DPCb) Au(CO)] + complex, where p back-donation is so strong as to invert the polarization of the p CO bonding orbitals in the C / O direction despite the formal positive charge on the ligand-metal fragment, making it the only example of a cationic gold(I) carbonyl complex with classical behavior (Dr CO > 0). An ad hoc study of CO in a uniform axial electric eld demonstrates that it is indeed the polarization of the p CO bonding orbitals, no matter how induced (whether by the coordination bond to M or by an electric eld), that drives direction and magnitude of the CO stretching response to the M-CO bond formation.
Regarding the fundamental question of what can be inferred on the nature of the M-CO bond from the analysis of Dr CO (and less reliably, due to mode coupling, Dn CO ) in metal carbonyl complexes, we conclude that the value of Dr CO quanties to an excellent extent the p back-donation component of the M-CO bond, since such component directly correlates with the p polarization. In particular, where CT p rCO=2 changes its sign (i.e. the polarization of p CO bonding orbitals changes direction determining whether the CO bond is weakened or strengthened), CT p back is approximately as high as the average extent of s donation among the complexes herein considered. This indicates that p back-donation is an important component also in the class of non-classical complexes, as those of gold(I) considered in this work.